The induced map between push outs in an exact infinity category

Question

Let $(\mathcal{C} , \mathcal{M} , \mathcal{E})$ be an exact $\infty$-category. (I am following the definition in Higher Segal Spaces $I$ by Dyckerhoff and Kapranov). Assume that $F$ is a cofiberation between two homotopy push out diagrams in $\mathcal{C}$. Is the induced morphism between push outs also a cofiberation?

Here is the link of the mentioned notes: https://arxiv.org/abs/1212.3563

The definition of an exact $\infty$-category can be found on page 116.

Thank you!